ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Two distinct one-dimensional modules over the generic Jordan Hall algebra

Example

Let $K = Q (t)$ and let $H_{J, K}$ be the generic Jordan Hall algebra. By The generic Jordan-quiver Hall algebra is the symmetric-function ring, $H_{J, K} = K [u_{1}, u_{2}, \dots]$ as an algebra, where $u_{r}$ is the generator corresponding to the class $u_{(1^{r})}$ . The assignments $ϵ_{0} (u_{r}) = 0 (r \geq 1), ϵ_{1} (u_{1}) = 1, ϵ_{1} (u_{r}) = 0 (r \geq 2)$ give two different one-dimensional Hall modules on the same $K$ -vector space $K$ , by $h \cdot z = ϵ_{i} (h) z$ .

Facts & Assumptions

Given: The generic Jordan Hall algebra over $K$ and the Hall-module definition.

[F1]

The generic Jordan-quiver Hall algebra is the symmetric-function ring: the generators $u_{r} = u_{(1^{r})}$ freely generate the generic Hall algebra as a polynomial algebra, so any assignment of their values in a commutative $K$ -algebra extends uniquely to a unital $K$ -algebra homomorphism.

[F2]

Modules over a Hall algebra are unital algebra representations: a unital representation $ρ : H \to {End}_{K} (M)$ defines a Hall-module action by $h \cdot m = ρ (h) m$ .

Verification

1.1givenF1construct

By [F1], assigning values to the free polynomial generators $u_{r}$ extends uniquely to a unital $K$ -algebra homomorphism for both $ϵ_{0}$ and $ϵ_{1}$ .

2.1givenF1F2step 1.1construct

For $i \in {0, 1}$ define $ρ_{i} (h) (z) = ϵ_{i} (h) z$ . Multiplicativity and unit preservation from step 1.1 make $ρ_{i}$ a unital representation, so [F2] gives a Hall-module action on $K$ .

3.1F1step 2.1algebra∎

The actions differ because $ρ_{0} (u_{1}) = 0$ while $ρ_{1} (u_{1}) = {id}_{K}$ . Thus the same one-dimensional vector space carries two different Hall-module structures.

Depends on

Used by

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Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §2.3 (standard reference, not scraped)